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Puzzle caches: Intersection of 2 (or 3) circles


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#1 Scott Dresser

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Posted 18 April 2006 - 01:00 PM

Without giving out spoilers one of the most common types of "math-based" puzzle caches I've run across (at least a dozen) are those based on the intersection of circles theme. You are given 2 or 3 points and distances from each point. The intersection of the circles is where the cache is, or clues to the next point, etc.

Dan pointed me to the proximity feature of EGPS and that helps in some cases it doesn't in others. Here's why:

1) EGPS distances are based on great circle calcs. See the discussion I started on the EGPS boards about other higher accuracy methods which other geocaching software tools use. It would be nice to have the ability to select the distance calculation method in EGPS -- people use both.

2) Proximity rings only display when the waypoint is visible in the map viewing pane. If you zoom in to plot the exact intersection the rings disappear. This limits how accurately you can pinpoint the intersection especially if the circle centers are far apart.

3) I think there is also some sort of bug with proximity rings when the distances involved are large (on the order hundreds or even thousands of miles). I was plotting the intersection of three circles the other day and I was using the proximity ring method in EGPS. As I zoomed in and out on a quick map the proximity rings would intersect at some zoom levels and not at others, like they were changing size based on zoom level. I haven't noticed this when the distances involved are on the order 10 miles but maybe its not noticeable.

The best solution would be the ability to draw regular shapes in EGPS, particularly circles given a center and a distance. If the distance algorithm used was selectable you would have a great solution for geocaching puzzles of this type.

-Scott

#2 Dan Foster

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Posted 18 April 2006 - 01:29 PM

Can you post some links to caches of this type on geocaching.com?

If there are spoilers involved, can you email me privately and just include the final geometry part of the puzzle (coordinates of the center points, plus distance and bearings)

#3 Scott Dresser

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Posted 18 April 2006 - 01:45 PM

Can you post some links to caches of this type on geocaching.com?


Here is a cache that uses this technique (its actually third in a series so you can look at the other two).

This particular one uses great circle calcs and involves long distances. It's pretty hard to solve using EGPS especially if you use the circle with the 2000+ mile radius. I approximated using EGPS and then used some software I wrote to get a final answer.

-Scott

#4 Dan Foster

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Posted 18 April 2006 - 02:57 PM

I made some changes that should make solving these caches much easier:

Proximity waypoints will always be drawn, even if the center point isn't on screen.
As you move a route waypoint using the Move Selected tool, the total distance will update in the Route List.

My approach to solving the problem:
1. Use proximity waypoints to determine a probable solution area.
2. Create a new waypoint "Solution" in the probable area.
3. Create three new routes, each starting at a geocache, and ending at Solution.
4. Switch to the Route list, and get the Total Distance column displayed on the left.
5. Switch to Quick Map for fast redrawing
6. Select Solution, and use the Move Selected tool to move it around.
7. Watch the route distances displayed on the left, and move Solution in the appropriate direction.
8. As you get closer to a solution, zoom in to fine-tune the results.

I "solved" the Traveling Cache #3 first, making changes to ExpertGPS as I went, and then "solved" Traveling Cache #1 in two minutes once I had the technique figured out. (Since I haven't actually driven out to find the caches yet, I won't claim victory. Also, ExpertGPS only shows route mileage to 2 decimal places, while the puzzle gives three. My "solution" may be off by 50 feet or so.)

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#5 Dan Foster

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Posted 18 April 2006 - 03:13 PM

Proximity waypoints will only display from offscreen when using Quick Map. On the topo or aerial photo maps, the UTM projection won't properly display a proximity waypoint from the other side of the country.

What other types of geocaching puzzles can be solved using ExpertGPS?

#6 Scott Dresser

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Posted 18 April 2006 - 05:34 PM

I made some changes that should make solving these caches much easier:

Proximity waypoints will always be drawn, even if the center point isn't on screen.
As you move a route waypoint using the Move Selected tool, the total distance will update in the Route List.


Great! Thanks Dan. But I'm wondering why you went to the trouble of plotting the routes. If the proximity feature works now such that you can zoom into the intersection, why not just plot the waypoints, plug in the proximities, zoom into the intersection of the proximity circles and drop a waypoint at the intersection? Would that work?

-Scott

#7 Scott Dresser

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Posted 18 April 2006 - 05:54 PM

What other types of geocaching puzzles can be solved using ExpertGPS?


Take a look at any cache by geometry. I think I've done them all and used EGPS to some extent to help me solve them. Most involve having some basic understanding of 2D shapes and projecting waypoints from a given reference. Bisecting a line in half is a common task, that can be done manually by waypoint projection. Being able to draw arbitrary closed shapes with specific interior angles and side lengths would be helpful in many of these puzzles.

-Scott

#8 Dan Foster

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Posted 18 April 2006 - 06:29 PM

I've made another change so that you can solve the puzzle using only proximity waypoints. The reason it didn't work before was that I was simply drawing a circle on the screen a fixed radius from the center point. That's a reasonable approximation if the world is flat (or if you're only dealing with a square mile or so of ground). For your 2000 mile radius circle, things look very different. The screenshot below shows the real shape of the points 2627.57 miles from the cache, as projected in the Quick Map.

From now on, ExpertGPS will draw the proximity "circle" by projecting 360 points, one for every degree of bearing around the center point, and then "connect the dots".


The three orange arcs are the proximity circles. The 3 purple lines are the routes I used to solve the puzzle earlier. You can see that they arrive at the same solution point, making the route approach no longer needed.

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#9 Tim Osborn

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Posted 18 April 2006 - 08:16 PM

I've made another change so that you can solve the puzzle using only proximity waypoints.


Dan:

I'm using 2.3.2b1 and these changes don't appear to be incorporated. So I guess you are holding them until the next Beta? How often do you think you will make new Beta's available for download?

Thanks

Tim Osborn

#10 Scott Dresser

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Posted 19 April 2006 - 07:19 AM

The three orange arcs are the proximity circles. The 3 purple lines are the routes I used to solve the puzzle earlier. You can see that they arrive at the same solution point, making the route approach no longer needed.


Perfect. I can't wait to try it out Dan!

The only thing I need now is the high accuracy distance mode for people who use that as part of their puzzles and you've got a great solution.

-Scott

#11 Dan Foster

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Posted 19 April 2006 - 07:30 AM

Perfect. I can't wait to try it out Dan!

The only thing I need now is the high accuracy distance mode for people who use that as part of their puzzles and you've got a great solution.

-Scott


If I take out the sphere-based distance formula and drop in the Vincenty algorithm, won't that cause problems when you go to solve puzzles that use great circle calculations? Do I really need to make this a program setting?

Do you have any examples of caches that require a high-accuracy calculation?

#12 Scott Dresser

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Posted 19 April 2006 - 01:08 PM

If I take out the sphere-based distance formula and drop in the Vincenty algorithm, won't that cause problems when you go to solve puzzles that use great circle calculations? Do I really need to make this a program setting?


Unfortunately yes. It would be nice to have a configuration option which specfies the mode because I've done caches which use both. You've seen some that use GC.

Do you have any examples of caches that require a high-accuracy calculation?


Here's one cache that requires high accuracy calcs.

Here's another even though it doesn't state it directly on this page.

In order to solve either of these caches you need to do some other work to get the center points to start from, they aren't given like the previous examples.

-Scott

#13 Dan Foster

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Posted 19 April 2006 - 02:29 PM

The new geocaching features are available. Please see ExpertGPS 2.3.2b2.

#14 Scott Dresser

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Posted 19 April 2006 - 05:56 PM

I just played around with the new beta version of EGPS and this is getting a lot easier! The only complaint I have is that proximity rings seem to get rounded to the nearest 10th of a mile -- that's not accurate enough to get an intersection that's useful to search for a cache in most cases. Did this change recently?

I'll play around with the route method to see if I can refine the answer and get something more accurate.

Is there any reason why the accuracy of the proximity ring couldn't be enhanced?

-Scott

#15 Tim Osborn

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Posted 20 April 2006 - 04:15 PM

The only complaint I have is that proximity rings seem to get rounded to the nearest 10th of a mile -- that's not accurate enough to get an intersection that's useful to search for a cache in most cases.
-Scott


Scott, I think this is primarily a display rounding issue. If you input .25 miles for proximity and click "OK" a proximity circle will be drawn. That circle is in fact 1320 feet in radius, one quarter of a mile. If you then edit the waypoint, you will see that the .25 miles you previously entered has now been rounded to .3 miles. But if you change nothing and then hit "OK" again, the proximity circle's radius will in fact change to .3 miles (1584 feet). It would probably be better if this "display rounding" didn't occur.

The above example only used hundredths of a mile, but you can also enter thousandths of a mile for proximity and the extra precision is understood by ExpertGPS. (e.g. the proximity cirlce for .251 miles is a little larger than .250 miles).

Tim Osborn

#16 Dan Foster

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Posted 20 April 2006 - 04:45 PM

That is exactly what is happening. ExpertGPS tries to avoid rounding errors showing up in the display ("13" looks better than "12.99999") so I was rounding off the proximity value to a single decimal digit. I've changed proximity to show up to three decimal places.

In general, I try to limit the precision of displayed numbers to meaningful values. "0.3 miles to destination" has some meaning to me when I'm hiking. "0.321 miles to destination" is just confusing - my internal pedometer isn't calibrated in thousandths of a mile.

If there are other areas where more display precision is needed for a particular task, please let me know.

#17 Tim Osborn

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Posted 20 April 2006 - 05:59 PM

If there are other areas where more display precision is needed for a particular task, please let me know.


Dan:

Like I indicated in my "Property Boundary Plotting" post in the Professional Uses Discussion, I would like to see more display precision and unit options for Areas, Distances and Bearings.

Am I correct in believing that ExpertGPS is correctly understanding and acting on high-precision data entry? For example, even though ExpertGPS currently only displays bearings to a full degree, Under "Project Waypoint" if I enter a bearing with 6 digits after the decimal point, does the program use that precision to accurately project the new waypoint?

You make a good point about the purpose of the user determining the desired level of precision in displayed numbers. Most consumer compasses only have markings every two degrees, so displaying bearings to the nearest full degree makes a lot of sense for hiking. On the other hand, if the purpose of the user is to plot the legal description of her ranch, display precision to the tenth of a second for bearings is desirable.

Might you consider making display precision a user setting under Preferences? The default precision could be set as it currently is, appropriate for casual uses like hiking, but if the purpose of the user required a higher level of display precision, she could increase it under Preferences.

Tim Osborn

#18 d.camell

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Posted 03 September 2007 - 10:07 PM

...

What other types of geocaching puzzles can be solved using ExpertGPS?

I know this is an old thread, but just wanted to add that I used expertGPS to make this cache and it worked out real easy.




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